The question says "x does not occur freely in A" (which means it is either the variable immediately following a quantifier or it is in the scope of some quantifier involving x). Let I be an interpetation with domain on D, and v be an I-assignment. Suppose . We wish to show that . I don't know if this is correct but I'm considering two cases depending on wether holds for all :

Case 1: holds for all . So so . Is this correct?

Case 2: There is some such that does not hold... And we assumed that there is a d such that . But how can I complete this part?

I greatly appreciate any help on how to prove this question...

Sep 21st 2011, 04:11 AM

emakarov

Re: Predicate Forms

The exact details depend on the definition of , but here are some remarks.

What is ? The formula A does not depend on x, so there is no sense in providing d to it.

The proof of equivalence should not require considering whether holds for all d. One should use the fact that if x is not free in A, then iff . So, if , then there is a such that . If , then and so . If , then and so .